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A new class of linear operators on \(\ell^2\) and Schur multipliers for them. (English) Zbl 1146.47017

Let \(B(\ell^2)\) be the set of all bounded linear operators on \(\ell^2\) and \(B_w(\ell^2)=\{A=(a_{ij})_{i,j\geq 1}: Ax\in\ell^2\) for all \(x\in\ell^2\) with \(| x_k| \downarrow 0\}\). Notice that \(B_w(\ell^2)\) contains unbounded operators. For two matrices \(A=(a_{ij})_{i,j\geq 1}\) and \(B=(b_{ij})_{i,j\geq 1}\), their Schur product is defined by \(A*B=(a_{ij}\cdot b_{ij})_{i,j\geq 1}\). Let \(X\) be either \(B(\ell^2)\) or \(B_w(\ell^2)\). The space of Schur multipliers on \(X\) is the set of all matrices \(M\) such that \(M*B\in X\) for all \(B\in X\). The main result of the paper says that \(B(\ell^2)\subset M(B_w(\ell^2))\). Let \(T\) be the set of all infinite Toeplitz matrices. It is also proved that \(M(B_w(\ell^2))\cap T\subset M(B(\ell^2))\cap T\).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
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